Copula Theory: an Introduction

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas. This version: September 14, 2009 1 Historical introduction The history of copulas may be said to begin with Fréchet [69]. He studied the following problem, which is stated here in dimension 2: given the distribution functions F1 and F2 of two random variables X1 and X2 defined on the same probability space ˆΩ ,F ,P, what can be said about the set Γ ˆF1,F2 of the bivariate d.f.’s whose marginals are F1 and F2? It is immediate to note that the set Γ ˆF1,F2, now called the Fréchet class of F1 and F2, is not empty since, if X1 and X2 are independent, then the distribution function ˆx1,x2(Fˆx1,x2 =F1ˆx1F2ˆx2 always belongs to Γ ˆF1,F2. But, it was not clear which the other elements of Γ ˆF1,F2 were. Preliminary studies about this problem were conducted in [64, 70, 88] (see also [30, 181] for a historical overview). But, in 1959, Sklar obtained the deepest result in this respect, by introducing the notion, and the name, of a copula, and proving the theorem that now bears his name [191]. In his own words [193]: Fabrizio Durante Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Austria e-mail: [email protected] Carlo Sempi Dipartimento di Matematica “Ennio De Giorgi” Università del Salento, Lecce, Italy e-mail: [email protected]